In hindsight we know that \(O(x)= (\partial ^2)^{\frac{n-2}{2}} \partial _a S^1_a(x)\)  [1]} There may be a proportionality constant between \(O(x)\)  and \((\partial ^2)^{\frac{n-2}{2}} \partial _a S_a(x)\) . We neglect this for simplicity because we are concerned only with the Lorentz transformation property of the operator \((\partial ^2)^{\frac{n-2}{2}} \partial _a S_a(x)\) . where \(S^1_a(x)\)  is the leading soft-photon operator which is a conformal primary with \((\Delta =1, l=1)\) . Now we also know from (REF ) that \(O(x)\)  must be a primary with \((\Delta =n, l=0)\) . This tells us that \(S^1_a(x)\)  must have a scalar descendent at level \((n-1)\)  which is also a primary. This is a non-trivial constraint because not every \((\Delta =1,l=1)\)  primary has a scalar primary descendent at level \((n-1)\) . We show that for \(n=4\)  this is possible only if the primary descendent \(\partial _a(\partial _aS^1_b-\partial _b S^1_a)\)  decouples from the \(S\) -matrix. This leads to a differential equation for the \(S\) -matrix element with an insertion of \(S^1_a(x)\) . The same phenomenon happens in the case of soft-graviton. Now we generalise this in the following way. We replace \(S^1_a(x)\)  with an arbitrary primary operator \(S^{\Delta }_{a_1...a_l}\)  with dimension \(\Delta \)  and integer spin \(l\)  which has a scalar descendent \(O(x)\)  at level \((n-\Delta )\) . This requires \(\Delta \le n\) . Now there are infinite number of such operators. Consider, for example, the \((\Delta =1,l=3)\)  primary operator \(S^1_{abc}(x)\) . This can be a leading soft spin-3 operator. This has a scalar descendent \(O(x) = (\partial ^2)^{\frac{n-4}{2}}\partial _a\partial _b\partial _c S^1_{abc}(x)\)  of dimension \(\Delta =n\) . We show that at least for \(n=4\) , \(O(x)=\partial _a\partial _b\partial _c S^1_{abc}(x)\)  cannot be a primary operator. So this is ruled out, i.e,  for \(n=4\)  or space-time dimension \(D=6\) , \(S^1_{abc}(x)\)  cannot contribute to the soft-charge of the \(U(1)\)  symmetry.
